Research

I’m interested in mathematical physics, and more specifically in math input for quantum mechanics, and for astrophysics. Things here are complicated, and disputed, and better take your gun, when you go out in town.

Here is my scientific profile at Google Scholar.

I’m currently taking a break from research, trying to reorganize the quantum group basics, via a series of books. Once done I intend to reborn as a physicist, of the worst kind, working on dark matter and dark energy.

Books

My main scientific project, since 2018, is that of writing a series of short math books, on quantum groups and their applications:

1. Linear algebra and group theory
2. Complex Hadamard matrices and applications (2019)
3. Free quantum groups and related topics (2019)
4. Quantum permutations and lattice models
5. Quantum isometries and noncommutative geometry (2020)
6. Methods of classical and free probability

Boy this is hard work. I’m doing it alone, hidden in the countryside, and with precious help from my cats. Check out my Instagram.

Blogs

As an alternative to the books, here is a series of blogs, on operator algebras, that I started writing during the 2020 lockdown:

1. Operator algebras
2. Quantum groups
3. Free probability
4. Subfactor theory
5. Noncommutative geometry

These blogs are now in their first version, short but readable. I will slowly improve, in the future, both the content and the layout.

Videos

As another alternative to the books, here are some series of tutorial videos, graduate course style, available at my Youtube channel:

“Introduction to quantum groups”
1. Operator algebras and noncommutative spaces –> 0:58
2. Compact and discrete quantum groups –> 0:52
3. Haar measure and Peter-Weyl theory –> 0:57
4. Tannakian duality, diagrams and easiness –> 0:58
5. Quantum permutations and quantum reflections –> 0:58
6. Orientability, toral subgroups and matrix models –> 0:53

“Introduction to Hadamard matrices”
1. The Hadamard conjecture –> 0:27
2. Complex Hadamard matrices –> 0:27
3. Deformed Hadamard matrices –> 0:25
4. Bistochastic Hadamard matrices
5. Almost Hadamard matrices
6. Hadamard matrix applications

Here are as well some more advanced series, for the most on quantum groups and their applications, minicourse/research style:

“Quantum group basics”
1. Compact quantum groups
2. Quantum permutation groups
3. Easy quantum groups
4. Transitive quantum groups

“Quantum groups and operator algebras”
1. Quantum permutations and Hadamard matrices
2. Quantum groups and subfactor theory
3. Quantum isometries and noncommutative geometry
4. Quantum reflections and free probability

Further videos at my channel, and in particular the “General mathematics” series there, containing short philosophical videos.

Papers

Here are my old math papers, written 1995-2019, all peer-reviewed and published, listed here according to their date of completion:

Early work

My MSc and PhD work, and follow-ups, mainly dealing with the basic theory of free quantum groups, and their applications:

B: On the polar decomposition of circular variables, 95
B: The representation theory of free orthogonal quantum groups, 96
B: The free unitary compact quantum group, 96
B: Representations of compact quantum groups and subfactors, 97
B: Hopf algebras and subfactors associated to vertex models, 97
B: Fusion rules for representations of compact quantum groups, 98
B: Symmetries of a generic coaction, 98
B: Subfactors associated to compact Kac algebras, 99
B: Compact Kac algebras and commuting squares, 99
B: Quantum groups and Fuss-Catalan algebras, 00
B: The planar algebra of a coaction, 02
B: Quantum automorphism groups of small metric spaces, 03
B: Quantum automorphism groups of homogeneous graphs, 03

Core theory

Fundamental quantum group work, done with Bichon, Bisch, Chenevier, Collins, Moroianu, Nicoara, Vergnioux:

BM: On the structure of quantum permutation groups, 04
BB: Free product formulae for quantum permutation groups, 05
BC: Integration over compact quantum groups, 05
BB: Quantum automorphisms of transitive graphs of order leq 11, 06
BB: Spectral measures of small index principal graphs, 06
BBC: Graphs having no quantum symmetry, 06
BC: Integration over quantum permutation groups, 06
BC: Integration over the Pauli quantum group, 06
BN: Quantum groups and Hadamard matrices, 06
BV: Growth estimates for discrete quantum groups, 06
BBC: Quantum permutation groups: a survey, 06

Further basics

Further fundamental quantum group work, done with Belinschi, Bichon, Capitaine, Collins, Speicher, Vergnioux:

BBC: The hyperoctahedral quantum group, 07
BB: Quantum groups acting on 4 points, 07
B: A note on free quantum groups, 07
BBCC: Free Bessel laws, 07
B: Cyclotomic expansion of exceptional spectral measures, 07
BV: Fusion rules for quantum reflection groups, 08
BB: Hopf images and inner faithful representations, 08
BS: Liberation of orthogonal Lie groups, 08

Intense easiness

Work done in 09, on easiness, with Bichon, Collins, Curran, Goswami, Schlenker, Speicher, Vergnioux, Zinn-Justin:

BCZ: Spectral analysis of the free orthogonal matrix, 09
BBS: Representations of quantum permutation algebras, 09
BCS: On orthogonal matrices maximizing the 1-norm, 09
BV: Invariants of the half-liberated orthogonal group, 09
BG: Quantum isometries and noncommutative spheres, 09
BCS: Classification results for easy quantum groups, 09
B: The orthogonal Weingarten formula in compact form, 09
BCS: De Finetti theorems for easy quantum groups, 09
BCS: Stochastic aspects of easy quantum groups, 09
BCS: On polynomial integrals over the orthogonal group, 09
B: The planar algebra of a fixed point subfactor, 09

New horizons

Exploring various related topics, with Bichon, Collins, Curran, Franz, Natale, Nechita, Schlenker, Skalski, Soltan:

BBC: Quantum automorphisms and free hypergeometric laws, 10
BC: Decomposition results for Gram matrix determinants, 10
BS: Two-parameter families of quantum symmetry groups, 10
BS: Combinatorial aspects of orthogonal group integrals, 10
BS: Quantum isometries of duals of free powers of cyclic groups, 10
BBN: Finite quantum groups and quantum permutation groups, 11
BN: Asymptotic laws of block-transposed Wishart matrices, 11
BBCC: A maximality result for orthogonal quantum groups, 11
B: Quantum permutations, Hadamard matrices and matrix models, 11
BSS: Noncommutative homogeneous spaces: the matrix case, 11
BS: Quantum symmetry groups of algebras with orthogonal filtrations, 11
BFS: Idempotent states and the inner linearity property, 11

Linear algebra

Basic Hadamard matrix work, and other, with Bhowmick, De Commer, Nechita, Schlenker, Skalski, Zyczkowski:

BBD: Quantum isometries and group dual subgroups, 12
BN: Block-modified Wishart matrices and free Poisson laws, 12
BNZ: Almost Hadamard matrices: general theory and examples, 12
B: The defect of generalized Fourier matrices, 12
BN: Almost Hadamard matrices: the case of arbitrary exponents, 12
BNS: Analytic aspects of the circulant Hadamard conjecture, 12
B: First order deformations of the Fourier matrix, 13
BS: The quantum algebra of partial Hadamard matrices, 13
BNS: Submatrices of Hadamard matrices: complementation results, 13
B: Counting results for thin Butson matrices, 13

Geometry and models

Work mostly on two topics, nocommutative geometry and matrix models, with Bichon, Meszaros, Nechita, Patri:

BB: Random walk questions for linear quantum groups, 14
B: The glow of Fourier matrices: universality and fluctuations, 14
B: Truncation and duality results for Hopf image algebras, 14
B: The algebraic structure of quantum partial isometries, 14
B: Liberations and twists of real and complex spheres, 14
B: Quantum isometries of noncommutative polygonal spheres, 15
B: A duality principle for noncommutative cubes and spheres, 15
B: Half-liberated manifolds and their quantum isometries, 15
B: Liberation theory for noncommutative homogeneous spaces, 15
BM: Uniqueness results for noncommutative spheres and projective spaces, 15
B: Quantum isometries, noncommutative spheres and related integrals, 16
BN: Flat matrix models for quantum permutation groups, 16
BP: Maximal torus theory for compact quantum groups, 16
B: Deformed Fourier models with formal parameters, 16
B: Quantum groups from stationary matrix models, 16
B: Weingarten integration over noncommutative homogeneous spaces, 16
BB: Matrix models for noncommutative algebraic manifolds, 16

The matrix

Heavy work done in 17, mostly on matrix models, with Bichon, Chirvasitu, Freslon, Nechita, Ozteke, Pittau:

BN: Almost Hadamard matrices with complex entries, 17
BB: Complex analogues of the half-classical geometry, 17
BF: Modelling questions for quantum permutations, 17
BC: Thoma type results for discrete quantum groups, 17
B: Complex Hadamard matrices with noncommutative entries, 17
B: Super-easy quantum groups: definition and examples, 17
BOP: Isolated partial Hadamard matrices and related topics, 17
B: Tannakian duality for affine homogeneous spaces, 17
BC: Quasi-flat representations of uniform groups and quantum groups, 17
B: Unitary easy quantum groups: geometric aspects, 17
B: Block-modified Wishart matrices: the easy case, 17
BC: Modelling questions for transitive quantum groups, 17
B: Higher transitive quantum groups: theory and models, 17

Final words

My last papers, consisting of a survey, two technical papers, and then a beginning of something new, that I eventually gave up with:

B: Quantum groups, from a functional analysis perspective, 18
B: Homogeneous quantum groups and their easiness level, 18
B: Higher orbitals of quizzy quantum group actions, 18
B: Quantum groups under very strong axioms, 19

Please note that some of these papers might contain errors. I’m fixing them now in my books, but the books might contain errors too.